\section{Flow}
\label{sec:flow}

Because the heavy-ion collisions are not always central, there will be an overlap region of the colliding nuclei.
In transverse plane the shape of this region is like ellipse. In the process of the scatterings, the spatial anisotropy will transform from initial state into final state. 
In order to quantify the momentum space anisotropy, we can expand the azimuthal distribution of the partons in the Fourier decompositions. 
The second Fourier coefficient is the largest contribution of shape of plane, we define it as $v_{2}$, the elliptic flow coefficient.
It is very important to study the how the ellipse transform in collisions. So we draw the v2 vesus number of collisions plots to reveal the relationship between them.
In figure.\ref{fig:v2_vs_n_all} and figure.\ref{fig:v2_vs_N_all} we'd like to discuss how dose the anisotropy develop. 
Here we have two kinds of different definitions of "number of collisions". The first one is the total number of collisons which a parton suffered before the collisoins sease, we call it "$N_{coll}$".
To be contract with $N_{coll}$, we defined $N^{*}_{Coll}$ which is the number of collisions the parton has in current state. So that we can better research on the medium states of the partons. 
In figure.\ref{fig:v2_vs_n_all}, the plot starts from 0 because the momentum space is isotropy in the initial state. After that the partons begin to expand in the direction of the shorter axis of the ellipsoid. 
So the value of $v_{2}$ is positive when $N^{*}_{Coll}>0$. However the partons with  $N^{*}_{Coll}$ comes from two composite. The first part are partons which are final state, for these part, the $v_{2}$ of them is positive, as we can see from figure.\ref{fig:v2_vs_N_all}. The remains are partons in intermediate states.
For these partons, they have not expanded along the shorter axis, so the value of their $v_{2}$ is negative. Combining these two factors, the $v_{2}$ value in figure.\ref{fig:v2_vs_n_all} is positive but quite small. 
As the $N^{*}_{Coll}>0$ goes larger, more composite comes from the final state partons, that is the reason why this plot rises with $N^{*}_{Coll}>0$.
In Figure.\ref{fig:v2_vs_N_all}, the $v_{2}$ here are all final state $v_{2}$ except for the partons with $N_{coll}=0$. They are in both initial state and final state because they did not collide with other partons. 
As we know the overlap region of the nuclei is along the longer axis, so that the partons have no scattering stay in shorter axis, that is why Figure.\ref{fig:v2_vs_N_all} start from a positive value.
For the similar reason, the longer a parton initiate in the momentum space, the more collisions that parton will have. Eventhough the collisions will result in the expansion along the direction of shorter axis. The intial shape of ellips also contribution to the $v_{2}$ value of the final state. 

\begin{figure}{}
  \begin{center}
  \subfigure[v2 of all partons vesus $N^{*}_{Coll}$]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_n_noptcut.pdf}
    \label{fig:v2_vs_n_all}
  }
  \subfigure[v2 of all partons vesus $N_{Coll}$]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_noptcut.pdf}
    \label{fig:v2_vs_N_all}
  }
  \caption{This 2 figures reveal the relationship between v2 and number of collisions. In the left panel, it's the v2 of partons in their mediem state. The x-axis here is the number of intermediate stage of collisions. That means a paron maybe recorded several times. For example, if a parton has 2 collisions before formations. The v2 of this parton will be recorded at $N_{Coll}^{*}=0,1,2$. This number will be defined as $N_{coll}^{*}$ to differ with $N_{coll}$ in the right panel, which means the total number of collisions of a certain parton. }
   \label{fig:v2_n_noptcut} 
   \end{center}
\end{figure}

Since we have got the "collision chain", we can get the number of collisions of every recorded parton, as well as the initial and final information of the parton. It's also a very important imformation of a parton.
In Figure \ref{fig:ncoll}, we show the number of collision distribution. The dot lines
mean distribution of intial partons and the active lines mean distribution of  the final partons. Partons with both high and
 low transverse momentum are showed seperately using red and black lines. The distribution of parton with low transverse momentum( $P_{T}<0.5GeV$ ) are similar. We know that the cross-section is indepandent to parton transverse momentum.
The difference between partons with higher transverse momentum and lower transverse momentum is the time when the parton was generated. The partons with lowwer transverse momentum appears earlier so that it have more chance to collide with other partons. The parton formation time from string-melting in AMPT was set as $tf=E_H/m_{T,H}^2$ where H represents the parent hadron of the parton. That also coincide with our conclusion.

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/Ncoll.pdf}
  \caption{The differencial distribution of number of collision in different $P_{T}$ region. The distributions including different initial (drawn in dashed lines) or final (drawn in solid lines) $P_{T}$ shreshold. The black lines mean low $P_{T}$ region, where $P_{T}<0.5GeV$ and red lines mean high $P_{T}$ region, where $P_{T}>3GeV$, for both initial and final $P_{T}$ cuts.}
  \label{fig:ncoll}
  \end{center}
\end{figure}

%
%\begin{figure}[!htbp]
%  \begin{center}
%  \subfigure[v2 vs $N_{coll}$ in different initial $P_{T}$ region]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_i_prof.pdf}
%    \label{fig:v2_vs_N}
%  }
%  \subfigure[v2 vs $N_{coll}$ in different final $P_{T}$ region]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_f_prof.pdf}
%    \label{fig:v2_vs_N_1}
%  }
%  \caption{v2 of Initial and final partons as a function of number of collision in different $P_{T}$ region. The left figure is for different initial $P_{T}$ regions. And the right one is for different final $P_{T}$ regions. In each figure,the dashed lines are v2 of partons intial state. The solided lines are v2 of partons final states.  }
%  \end{center}
%\end{figure}
%

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/v2_vs_N.pdf}
  \caption{v2 of partons in their final states as a function of total number of collision in different final $P_{T}$ region. The black line is v2 distribution of all the partons. And red, green and blue lines show partons with low, medium and higt final state energy respectively.}
  \label{fig:v2_vs_N}
  \end{center}
\end{figure}


%\begin{figure}[!htbp]
%  \begin{center}
%  \subfigure[v2 vs different final $P_{T}$]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_i_prof.pdf}
%    \label{fig:v2_vs_pt}
%  }
%  \subfigure[v2 vs different final $P_{T}$]{
%    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_pt_f_prof.pdf}
%    \label{fig:v2_vs_pt_1}
%  }
%  \caption{These figures show relationship of v2 and parton $P_{T}$, the left one is for parton initial  $P_{T}$ and right figure is for final $P_{T}$. As last figure, the dashed lines are v2 of partons intial state. The solided lines are v2 of partons final states.}
%  \end{center}
%\end{figure}
\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/v2_vs_pt.pdf}
  \caption{These figure shows relationship of v2 and parton final $P_{T}$, The black line is the v2 distribution of all partons. In order to bettter analysis it, we split the sample based on the number of collisions. v2 vesus $P_{T}$ of different number of collisions are drawn with different color lines.}
  \label{fig:v2_vs_pt}
  \end{center}
\end{figure}



